Finite Group Databasebeta

← #42122615

Subgroups — #42122615

Order 27712.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28357266C211yesmaximal, characteristic, complemented
2#28357265C211yesmaximal, characteristic, complemented
3#28357248C211yesmaximal, characteristic, complemented
4#28357247C212yesmaximal, complemented
5#28357244C212yesmaximal, complemented
6#28357241C212yesmaximal, complemented
7#28357240C211yesmaximal, characteristic, complemented
8C2×#20294082C211yesmaximal, characteristic, complemented
9D8×D433C212yesmaximal, complemented
10#28357228C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C433C211yesmaximal, characteristic, complemented
12#20294098C2212yescomplemented
13#20294097—21no—
14#20294097C2211yescharacteristic
15#20294097C2211yescharacteristic, complemented
16#20294096—21no—
17#20294096C2212yescomplemented
18C2×D1732—21no—
19C2×D1732C2211yescharacteristic, complemented
20C2×C22⋊D433—22no—
21D4×D433—21no—
22D4×D433—22no—
23D4×D433C2212yescomplemented
24D4×D433C2212yescomplemented
25#20294087C2211yescharacteristic, complemented
26#20294086C2212yescomplemented
27#20294085C2211yescharacteristic, complemented
28#20294084C2211yescharacteristic, complemented
29D3464C2212yescomplemented
30#20294082C2211yescharacteristic, complemented
31#20294082C2212yescomplemented
32#20294082C2212yescomplemented
33#20294080C2212yescomplemented
34C2×#16812358C2211yescharacteristic, complemented
35C8×D433C2212yescomplemented
36#41×C433C2211yescharacteristic, complemented
37C2×D4×C433C2211yescharacteristic, complemented
38SD16×C433C2212yescomplemented
39D8×C433C2212yescomplemented
40M16×C433C2211yescharacteristic, complemented
41C22×D433—21no—
42C22×D433—22no—
43C22×D433—22no—
44Dic866D411yescharacteristic
45Dic866C2311yescharacteristic
46D1732—22no—
47D1732—21no—
48D1732D411yescharacteristic
49D1732C2312yes—
50D1732C2311yescharacteristic
51C22⋊D433—21no—
52C22⋊D433—22no—
53C22⋊D433—22no—
54C22⋊D433—22no—
55C22⋊D433D412yes—
56C2×Dic433—22no—
57C4×D433—21no—
58C4×D433—22no—
59C4×D433C2312yes—
60#16812358C2311yescharacteristic
61#16812358C2311yescharacteristic
62C23×C433—21no—
63Q8×C433C2311yescharacteristic
64D4×C433—21no—
65D4×C433—21no—
66D4×C433C2311yescharacteristic
67D4×C433C2312yes—
68C2×C4×C433—21no—
69C2×C4×C433C2311yescharacteristic
70C8×C433C2312yes—
71C22×C433—21no—
72C22×C433—22no—
73C22×C433—81no—
74C22×C433C2×D411yescharacteristic
75C2×D433—21no—
76C2×D433—22no—
77C2×D433—82no—
78C2×D433—82no—
79C2×D433—81no—
80C2×D433C2×D412yes—
81C4×C433—21no—
82C4×C433C2×D411yescharacteristic
83C4×C433C2411yescharacteristic
84Dic433—21no—
85Dic433C2×D412yes—
86C2×C433—21no—
87C2×C433—41no—
88C2×C433—42no—
89C2×C433C22×D411yescharacteristic
90D433—22no—
91D433—41no—
92D433—42no—
93C433#57611yescharacteristic, complemented, Sylow
94#576—4331nomaximal, Sylow
95H(5,𝔽2)—4331no—
96H(5,𝔽2)—4331no—
97AGL(1,ℤ/8)—4332no—
98AGL(1,ℤ/8)—4332no—
99AGL(1,ℤ/8)—4331no—
100AGL(1,ℤ/8)D43311yescharacteristic, complemented
101#135—4331no—
102#135—4332no—
103C2×D8—4331no—
104C2×D8—4332no—
105#131—4331no—
106#41—8661no—
107#41—8661no—
108#41—4332no—
109#41—4332no—
110#41—4331no—
111#41—4331no—
112#41C2×D43311yescharacteristic, complemented
113C2×D4—8661no—
114C2×D4—8662no—
115C2×D4—8661no—
116C2×D4—8662no—
117C2×D4—4332no—
118C2×D4—4332no—
119C2×D4—4331no—
120C2×D4C2×D43311yescharacteristic, complemented
121Q16—4331no—
122SD16—4331no—
123SD16—4331no—
124SD16—4332no—
125SD16C2×D43312yescomplemented
126D8—4332no—
127D8—4331no—
128D8—4332no—
129D8—4332no—
130D8C2×D43312yescomplemented
131M16—4332no—
132M16C2×D43311yescharacteristic, complemented
133C2×C8—4331no—
134C2×C8—4332no—
135C23—21no—
136C23—8662no—
137C23—8662no—
138C23—8661no—
139Q8—4331no—
140Q8—4331no—
141Q8C22×D43311yescharacteristic
142D4—21no—
143D4—21no—
144D4—8662no—
145D4—8662no—
146D4—8661no—
147D4—8662no—
148D4—8662no—
149D4—8661no—
150D4—4332no—
151D4—4331no—
152D4—4332no—
153D4—4331no—
154D4C22×D43312yes—
155D4C22×D43311yescharacteristic
156C2×C4—21no—
157C2×C4—8662no—
158C2×C4—8662no—
159C2×C4—8661no—
160C2×C4—4332no—
161C2×C4C22×D43311yescharacteristic
162C8—4331no—
163C8—4331no—
164C8C22×D43312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—8662no—
169C22—8661no—
170C22—34641no—
171C22—34642no—
172C22—34642no—
173C22—4332no—
174C22D4×D43311yescharacteristic
175C4—21no—
176C4—8661no—
177C4—4332no—
178C4D4×D43311yescharacteristic
179C4C23×D43311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8662no—
184C2—17322no—
185C2—17321no—
186C2C2×D4×D43311yescharacteristic
subgroupG/H
centerC2C2×D4×D433
commutator, cosocleC4×C433C24
FrattiniC4C23×D433
FittingAGL(1,ℤ/8)×C433C2
socleC2×C433C22×D4
hypercenterAGL(1,ℤ/8)D433
hypocenterC433#576
LCSG ▷ C4×C433 ▷ C2×C433 ▷ C433
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #28357266C2semidirect
2. #28357265C2semidirect
3. #28357248C2semidirect
4. #28357247C2semidirect
5. #28357244C2semidirect
6. #28357241C2semidirect
7. #28357240C2semidirect
8. C2×#20294082C2semidirect
9. D8×D433C2semidirect
10. #28357228C2semidirect
11. AGL(1,ℤ/8)×C433C2semidirect
12. #20294098C22semidirect
15. #20294097C22semidirect
17. #20294096C22semidirect
19. C2×D1732C22semidirect
23. D4×D433C22semidirect
24. D4×D433C22semidirect
25. #20294087C22semidirect
26. #20294086C22semidirect
27. #20294085C22semidirect
28. #20294084C22semidirect
29. D3464C22semidirect
30. #20294082C22semidirect
31. #20294082C22semidirect
32. #20294082C22semidirect
33. #20294080C22semidirect
34. C2×#16812358C22semidirect
35. C8×D433C22semidirect
36. #41×C433C22semidirect
37. C2×D4×C433C22semidirect
38. SD16×C433C22semidirect
39. D8×C433C22semidirect
40. M16×C433C22semidirect
93. C433#576semidirect
100. AGL(1,ℤ/8)D433semidirect
112. #41C2×D433semidirect
120. C2×D4C2×D433semidirect
125. SD16C2×D433semidirect
130. D8C2×D433semidirect
132. M16C2×D433semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#283572662
#283572652
#283572482
#283572472
#283572442
#283572412
#283572402
C2×#202940822
D8×D4332
#283572282
AGL(1,ℤ/8)×C4332
#576433